Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(3060\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \cdot 17 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.45900.4 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.340.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-15}, \sqrt{51})\) |
Defining polynomial
$f(x)$ | $=$ |
\( x^{4} + 12x^{2} + 51 \)
|
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 9 + 26\cdot 79 + 52\cdot 79^{2} + 31\cdot 79^{3} + 15\cdot 79^{4} +O(79^{5})\)
|
$r_{ 2 }$ | $=$ |
\( 12 + 10\cdot 79 + 4\cdot 79^{2} + 54\cdot 79^{3} + 34\cdot 79^{4} +O(79^{5})\)
|
$r_{ 3 }$ | $=$ |
\( 67 + 68\cdot 79 + 74\cdot 79^{2} + 24\cdot 79^{3} + 44\cdot 79^{4} +O(79^{5})\)
|
$r_{ 4 }$ | $=$ |
\( 70 + 52\cdot 79 + 26\cdot 79^{2} + 47\cdot 79^{3} + 63\cdot 79^{4} +O(79^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 4 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 4 }$ | Character value | Complex conjugation |
$1$ | $1$ | $()$ | $2$ | |
$1$ | $2$ | $(1,4)(2,3)$ | $-2$ | |
$2$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
$2$ | $2$ | $(1,4)$ | $0$ | |
$2$ | $4$ | $(1,3,4,2)$ | $0$ |